Introducing Game Theory: A Graphic Guide (Graphic Guides)

by Ivan Pastine

There are possible ways to avoid coordination failure in games with more than one equilibrium …

In environments with multiple equilibria, players may coordinate their expectation on one equilibrium using social norms.

While the Battle of Sexes Game does not give conditions under which societies evolve into patriarchy (a society structured around male advantage), it does give an insight into a potential benefit of gender-based dominance. This may be one of the reasons why it is so difficult to move society to a fairer system.

In games with multiple equilibria, if a social norm is not present, players may use a coordination device, a shared observation or common history to help coordinate expectations on the same equilibrium.

no matter how healthy the financial state of a bank may be, any bank will go under if faced with a bank run (where everybody tries to withdraw their money at the same time).

The belief that there will be a bank run is a self-fulfilling expectation: the expectation itself causes the bank run.

Since there are always multiple equilibria in banking, people’s expectations about what will happen determine the outcome. Even positive statements or actions by bankers or policymakers can backfire if people take them as a sign of weakness.

pure-strategy Nash equilibrium. This is an equilibrium in which players pick a particular choice with certainty.

zero-sum game: if one person wins, the other loses.

it has no equilibrium where players behave predictably.

players try to be unpredictable; the game does not have a pure-strategy Nash equilibrium.

mixed-strategy Nash equilibrium.

in equilibrium the players randomize over possible pure strategies:

mixed strategies of the players need to be best responses to each other to form a Nash equilibrium.

speculative attack

Mixed-strategy Nash equilibrium is also interesting in environments with multiple pure-strategy Nash equilibria where each player prefers a different equilibrium outcome.

An economic application of the Chicken Game is the Exit Game.

the business is indifferent if its expected profit from “stay” is equal to its expected profit from “exit”.

If the expected profit from one action were higher than the other, the business would prefer the action with the higher expected profit, and it would choose that action with certainty. In equilibrium, there is randomization and hence uncertainty about a store’s action only when the store is indifferent, when its expected profit is the same from either action.

war of attrition.

the mixed-strategy Nash equilibrium is often the most intuitive since it can capture uncertainty in these environments.

critics of mixed strategies argue that randomization is not a reasonable description of human behaviour.

One powerful defence of mixed strategies is the “purification” interpretation of mixed-strategy Nash equilibrium.

even if players play pure strategies, if they are slightly uncertain about each other’s payoffs, from the outside they will seem as if they are randomizing between actions.

if the players are almost, but not quite, certain about each other’s payoffs, from the individual’s point of view the other player’s chance of choosing a particular action is exactly the probability we get in a mixed-strategy Nash equilibrium without uncertainty about payoffs.

when players collude in a Prisoners’ Dilemma type of situation, we need to move beyond one-shot games (where players play the game only once and then the game ends) and to start thinking about more realistic settings with repeated interaction, where players play the same game again and again.

To find the equilibrium of the game with repeated interaction, we first predict the equilibrium of the game in the last round. And then we reason what the equilibrium would be in the first round. This line of reasoning is called backward induction.

even if the Prisoners’ Dilemma Game were repeated over many rounds, we would never observe cooperation in any round as long as the game has a certain final round. Backward induction unravels the game from the last round.

game has an infinite horizon, which means that the game is repeated forever. With an infinite horizon, backward induction does not unravel cooperation from the last round, since there is no certain last round.

The first condition for cooperation to be an equilibrium outcome is that players’ strategies have an element of punishment for past bad behaviour (non-cooperative actions). To avoid future punishment, players may choose to be cooperative.

Both players playing the grim strategy can be a Nash equilibrium in a repeated Prisoners’ Dilemma

if players are patient enough (if they are able to resist the temptation of a high payoff today in order to be able to collect cooperative payoffs in the future). In this case, punishment for defection can deter the players from non-cooperative actions.

if players are impatient, they will be tempted to defect today despite the punishment in the future. Knowing this, the rival would not behave cooperatively in the first place. So, with impatient pl...

When players are patient, for the threat of punishment to be a deterrent to defection, the threat needs to be credible

Hence, if collusion breaks down, both players have an incentive to renegotiate, ignore the deviation and simply start colluding all over again.

However, if players expect renegotiation to take time, then the threat can have a deterrent effect and produce a collusive outcome in equilibrium.

Cooperative outcomes were often recorded as long as the end of the game was not in sight. But as time was running out and the end of the game drew near, players started to defect and mutual coordination broke down.

think of people or animals as being socially or genetically programmed to engage in certain behaviours, which may or may not be based on reason.

importance of evolutionary stability, which examines which types of behavioural patterns are likely to survive evolutionary forces.

the game assumes that there are two types of animals in a species: “hawk” and “dove”. The hawk-type fights if necessary when competing for a prize, such as a mating opportunity or a scarce resource. The dove-type makes an aggressive display, but falls short of nonceremonial physical conflict.

animal’s evolutionary fitness. Access to the contested prize improves the animal’s prospects of reproduction or survival (whether the prize is a mating opportunity or scarce resources). The higher the payoff, the better evolutionary fitness the animal has.

When the cost of conflict is less than the value of the prize, animals that behave aggressively do better than less aggressive animals, whichever type they get matched with.

Evolutionary forces do not necessarily lead to the best outcome for a species. Competition for scarce resources often means that individual benefits and group benefits are in opposition. Whenever this is the case, the species will evolve to maximize individual benefits at the expense of group benefits.

evolutionarily stable equilibrium. It is an equilibrium which is stable in the sense that if we add a small number of animals with different conditioning, evolutionary forces will eventually restore the equilibrium.

sequential-move games. Most board games, such as chess, have alternating sequential moves.

Sequential-move games are dynamic,

Players conjecture what the other players would do in response to their possible choices, and then work backwards from the end of the game in order to decide what to do.

the extensive-form representation. This is also called the game tree.

The extensive form introduces the order of choices through the use of decision nodes, dots that represent a point at which a decision can be made.

think of the game from that point on as a game in itself. This is known as a subgame.